We present eight proofs of ![E[e^{itZ}] = e^{-t^2/2}](https://s0.wp.com/latex.php?latex=E%5Be%5E%7BitZ%7D%5D+%3D+e%5E%7B-t%5E2%2F2%7D&bg=ffffff&fg=111111&s=0&c=20201002)
where 
is a standard normal random variable. This fact is more or less equivalent to saying that the Gaussian is its own Fourier transform. We start with proofs that are rather elementary, and move on to more sophisticated proofs.
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